Near Wall Treatment and y+ Estimation

We must develop a mesh or grid that can precisely estimate the velocity gradient across the boundary layer from the perspective of CFD. The first cell from the wall should preferably be contained within the extremely thin viscous sub-layer in turbulent flows. Although this might be feasible in some flow scenarios, it cannot be achieved in the case of complex flows in complex geometries since it would necessitate a very fine mesh resolution close to the wall, which would significantly lengthen the time needed to solve the issue. A wall function is created to address this requirement, allowing the use of a "relatively" larger mesh close to the wall. Depending on the area of the boundary, right now.

What is y+ in CFD?

The unmeasurable variable Y-plus is a distance with no dimensions and is used to express the distance from the wall to the centre of the first grid cell. In other words, Y-plus makes the vertical distance between the centre of the element and the wall dimensionless; using this value, the proportionality of the height of the grid element to the borders of the wall is assessed.

On the other side, Y-plus represents the ratio of laminar to turbulent flow effects in a cell. The flow of the cell is laminar if Y-plus is low. If Y-plus is too tiny, the flow in the cell will be laminar slow, and the wall functions won't work because if Y-plus is too large, the current in that cell will be turbulent.

How do we resolve the gradient near the wall? 

As we approach the wall, the velocity decreases, leading one point to a laminar layer (Viscous sublayer). Here it becomes crucial to understand the velocity gradient as they are quite large. To evaluate these values numerically, we need a mesh sufficiently fine to accommodate the cells in the viscous sublayer. As this is done, the mesh will be able to resolve the velocity gradient near the wall - which happens to be linear. 

From the image above, we can see the sharpness of the velocity gradient near the wall. Hence, it becomes necessary to have a finer mesh near the wall to resolve the gradient accurately.

Now we have established that we need a fine mesh near the wall to evaluate the velocity, temperature, etc, gradient accurately. But, does this mean we have to make a fine mesh even when we are not interested in these quantities? If not, what can we do to have fairly accurate results near the wall without compromising on the overall solution? The answer to this question is a Wall function.

What is a Wall Function? 

A wall function is a mathematical relationship approximating the wall's effects. Here, a nonlinear function is used to understand the gradient from the wall to the cell centroid.

This cell is created in the log-law region, and we can use empirical correlations to bridge wall conditions to the log-law layer. This negates the need to have a fine mesh near the wall. 

What is this function?

The plot above is a relationship established via experiments and Direct numerical simulations. Here, we can see how the u+ (Velocity)  and y+ (distance normal to the wall)are related. Now, to approximate these results via an empirical relation, we need to establish some functions, what are they. These functions will fit the curve above. The result of this is shown in the image below.

To understand these functions, we need to define some dimensionless numbers. 

• For a standard wall function,
• Y+  = y*ut u+= Uut
• Here, ut is the reference velocity based on wall shear stress near the wall ut = (w/)½
• Here, 
• U+ = Y+ y+ < 5 
• U+ = 1log(Ey+) 30<y+<200
• Note: We plot with dimensional quantities because we need the results to be universal. 

A point to note here is that there is no function to represent the buffer layer as the 2 curves do not fit this layer. What can be done in this regard?

To counter this Spalding wall function is used. But there are more wall functions that are used to counter this issue. 

The images below show the mesh besides the velocity profile for a better understanding of how to structure a mesh based on the approach that needs to be used. Here, in the mesh resolving approach, the first cell needs to be placed in the viscous sublayer with y+ 1-5. In this approach to capture the near wall effects, it is better to place 8-10 cells (minimum 5)  in the viscous sublayer with an appropriate growth rate. 

Whereas, If you are not interested in resolving the boundary layer up to the viscous sub-layer, you can use wall functions. In terms of y+, wall functions will model everything below y+ < 30 or the target y+ value. This approach uses coarser meshes, but you should be aware of the limitations of the wall functions. You will need to cluster at least 5 to 10 layers close to the walls in order to resolve the profiles (U, k, epsilon, Reynolds stresses and so on). As a general rule, when using wall functions, the first cell center should be located above y+ > 40-50 and below (boundary layer thickness).

One can see the wall modelling approach using wall functions is a better option as it requires less computational resources. However, when it comes to having a flow with separation or heat transfer problems, it is better to resolve the mesh as it is a more accurate approach. When using the wall resolving approach, try to get an average y+ value close to 1 or lower. Values of y+ lower than 0.1 will not give a large improvement. Pushing the mesh to values of y+ below 0.1 can result in low-quality meshes for industrial applications. It is a common agreement that the upper limit of the viscous sublayer is five. When placing the first cell center, you can go as high as six without losing too much accuracy. But ideally, you should aim for a y+ value of 1-2.